\(\int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx\) [793]

Optimal result

Integrand size = 22, antiderivative size = 144 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \]

[Out]

3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a/c-5/3*(1+1/a/x)^(1/2)/a/c/(1-1/a/x)^(3/2)+x*(1+1/a/x)^(1/2)/c/(1-
1/a/x)^(3/2)-14/3*(1+1/a/x)^(1/2)/a/c/(1-1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6329, 101, 157, 12, 94, 214} \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c}+\frac {x \sqrt {\frac {1}{a x}+1}}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {\frac {1}{a x}+1}}{3 a c \sqrt {1-\frac {1}{a x}}}-\frac {5 \sqrt {\frac {1}{a x}+1}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}} \]

[In]

Int[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]

[Out]

(-5*Sqrt[1 + 1/(a*x)])/(3*a*c*(1 - 1/(a*x))^(3/2)) - (14*Sqrt[1 + 1/(a*x)])/(3*a*c*Sqrt[1 - 1/(a*x)]) + (Sqrt[
1 + 1/(a*x)]*x)/(c*(1 - 1/(a*x))^(3/2)) + (3*ArcTanh[Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]])/(a*c)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 94

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 6329

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 - x/a)^(p
- n/2)*((1 + x/a)^(p + n/2)/x^2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Integ
erQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegersQ[2*p, p + n/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \left (1-\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {\frac {3}{a}+\frac {2 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {a \text {Subst}\left (\int \frac {-\frac {9}{a^2}-\frac {5 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c} \\ & = -\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {a^2 \text {Subst}\left (\int \frac {9}{a^3 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c} \\ & = -\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = -\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2 c} \\ & = -\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.48 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {\frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (14-19 a x+3 a^2 x^2\right )}{(-1+a x)^2}+\frac {9 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \]

[In]

Integrate[E^(3*ArcCoth[a*x])/(c - c/(a^2*x^2)),x]

[Out]

((Sqrt[1 - 1/(a^2*x^2)]*x*(14 - 19*a*x + 3*a^2*x^2))/(-1 + a*x)^2 + (9*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x])/a)/
(3*c)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {a x -1}{a c \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}-\frac {2 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}-\frac {13 \sqrt {\left (x -\frac {1}{a}\right )^{2} a^{2}+2 \left (x -\frac {1}{a}\right ) a}}{3 a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{c \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\)	\(185\)
default	\(-\frac {-9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-9 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x +27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} x^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-27 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -27 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}+9 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )}{3 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(346\)

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2),x,method=_RETURNVERBOSE)

[Out]

1/a*(a*x-1)/c/((a*x-1)/(a*x+1))^(1/2)+(3/a^2*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)-2/3/a^5/(x-1/
a)^2*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2)-13/3/a^4/(x-1/a)*((x-1/a)^2*a^2+2*(x-1/a)*a)^(1/2))*a^2/c/((a*x-1)/(a*x
+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)/(a*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2),x, algorithm="fricas")

[Out]

1/3*(9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 9*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(
a*x + 1)) - 1) + (3*a^3*x^3 - 16*a^2*x^2 - 5*a*x + 14)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c
)

Sympy [F]

\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {a^{2} \int \frac {x^{2}}{\frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]

[In]

integrate(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2),x)

[Out]

a**2*Integral(x**2/(a**3*x**3*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(
a*x + 1))/(a*x + 1) - a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x
+ 1)), x)/c

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {1}{3} \, a {\left (\frac {\frac {11 \, {\left (a x - 1\right )}}{a x + 1} - \frac {18 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2),x, algorithm="maxima")

[Out]

1/3*a*((11*(a*x - 1)/(a*x + 1) - 18*(a*x - 1)^2/(a*x + 1)^2 + 1)/(a^2*c*((a*x - 1)/(a*x + 1))^(5/2) - a^2*c*((
a*x - 1)/(a*x + 1))^(3/2)) + 9*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 9*log(sqrt((a*x - 1)/(a*x + 1)) -
1)/(a^2*c))

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)/(c-c/a^2/x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx=\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {11\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]

[In]

int(1/((c - c/(a^2*x^2))*((a*x - 1)/(a*x + 1))^(3/2)),x)

[Out]

(6*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c) - ((11*(a*x - 1))/(3*(a*x + 1)) - (6*(a*x - 1)^2)/(a*x + 1)^2 + 1
/3)/(a*c*((a*x - 1)/(a*x + 1))^(3/2) - a*c*((a*x - 1)/(a*x + 1))^(5/2))